Question 1

Solve the system. $\left(\begin{array}{rl}
4 x+3 y-4 z & =-15 \
3 x-3 y+4 z & =43 \
5 y+z & =-21
\end{array}\right)$

Accepted Answer

The answer of Solve the system. \(\left(\begin{array}{rl}
4 x+3 y-4 z...

Question 2

A video store rents new release movies for $5 and favorites for $2.55. One day the number of new release movies rented was twice the number of favorites. If the total income for that day was $815.75, how many movies of each kind were rented?

Accepted Answer

A)  140 new release, 70 favorites 
B)  150 new release, 75 favorites 
C)  70 new release, 140 favorites 
D)  65 new release, 130 favorites 
E)  130 new release, 65 favorites 
A)  140 new release, 70 favorites 
B)  150 new release, 75 favorites 
C)  70 new release, 140 favorites 
D)  65 new release, 130 favorites 
E)  130 new release, 65 favorites

Question 3

The augmented matrix of a system of equations is a matrix of the coefficients of the equations.

Accepted Answer

A) True 
 B)False

Question 4

Match each step with step-by-step description of how you would solve the system. $$\left( \begin{array} { c } 
2 x - y + 3 z = 24 \
x - 2 y - z = 1 \
3 x + 5 y + 8 z = 47
\end{array} \right)$$
-Finding x , y , z . x = 4, y = -1 , z = 5.

Accepted Answer

A)  1st step 
B)  2nd step 
A)  1st step 
B)  2nd step

Question 5

Solve the system by using the elimination-by-addition method. $$\left( \begin{array} { l } 
3 x &+ &8 y &= &3 \
5 x& - &8 y& = &1
\end{array} \right)$$ If the equations of the system are dependent, or if a system is inconsistent, so indicate. In those cases enter dependent or inconsistent . Otherwise, enter your answer as an ordered pair ( x , y ).

Accepted Answer

The answer of Solve the system by using the elimination-by-addition...

Question 6

Evaluate $2 \times 2$ determinant. $$\left|\begin{array} { c c } 
\frac { 1 } { 4 } & \frac { 4 } { 15 } \
\frac { 15 } { 16 } & - \frac { 1 } { 15 }
\end{array} \right|$$

Accepted Answer

A)  $- \frac { 4 } { 5 }$ 
B)  $- \frac { 4 } { 15 }$ 
C)  $- \frac { 1 } { 4 }$ 
A)  $- \frac { 4 } { 5 }$ 
B)  $- \frac { 4 } { 15 }$ 
C)  $- \frac { 1 } { 4 }$

Question 7

Solve the system by the substitution or elimination method. $$\left( \begin{array} { r l } 
7 x ^ { 2 } + y ^ { 2 } & = 343 \
x ^ { 2 } + y ^ { 2 } & = 49
\end{array} \right)$$

Accepted Answer

A)  ( $-$ 7, 0), (7, 0) 
B)  ( $-$ 7, 0), (0, 7) 
C)  (0, $-$ 7), (7, 0) 
D)  (0, $-$ 7), (0, 7) 
A)  ( $-$ 7, 0), (7, 0) 
B)  ( $-$ 7, 0), (0, 7) 
C)  (0, $-$ 7), (7, 0) 
D)  (0, $-$ 7), (0, 7)

Question 8

Graphs of two hyperbolas will have have no points of intersection.

Accepted Answer

A) True 
 B)False

Question 9

Use Cramer s rule to find the solution set for the system. $$\left( \begin{array} { r l } 
4 x - y + 5 z & = 31 \
5 y + z & = 23 \
x - 4 y - z & = - 14
\end{array} \right)$$ If the equations of the system are dependent, or if a system is inconsistent, so indicate. In those cases enter dependent or inconsistent .

Accepted Answer

The answer of Use Cramer s rule to find the...

Question 10

The matrix $\left( \begin{array} { l } 
3 x + y - z = 6 \
2 y + 3 z = 4 \
4 z = 10
\end{array} \right)$ is in echelon form.

Accepted Answer

A) True 
 B)False

Question 11

Solve the system. $$\left( \begin{array} { r l } 
x + 2 y - z & = 9 \
x + 3 y + 2 z & = 11 \
- x - 3 y - 2 z & = - 11
\end{array} \right)$$

Accepted Answer

A)  $( 7 k + 5 , - 3 k + 2 , k )$ , $k$ is a real number 
B)  $( 6 k + 6 , - 2 k + 2 , k )$ , $k$ is a real number 
C)  $( 7 k + 5 , - 2 k + 2 , k )$ , $k$ is a real number 
D)  $( 6 k + 6 , - 3 k + 2 , k )$ , $k$ is a real number 
E)  $( 6 k + 5 , - 3 k + 2 , k )$ , $k$ is a real number 
A)  $( 7 k + 5 , - 3 k + 2 , k )$ , $k$ is a real number 
B)  $( 6 k + 6 , - 2 k + 2 , k )$ , $k$ is a real number 
C)  $( 7 k + 5 , - 2 k + 2 , k )$ , $k$ is a real number 
D)  $( 6 k + 6 , - 3 k + 2 , k )$ , $k$ is a real number 
E)  $( 6 k + 5 , - 3 k + 2 , k )$ , $k$ is a real number

Question 12

Use Cramer s rule to find the solution set for the system. $$\left( \begin{array} { l } 
y = 2 x - 7 \
4 x - 5 y = 5
\end{array} \right)$$ If the equations of the system are dependent, or if a system is inconsistent, so indicate. In those cases enter dependent or inconsistent . Otherwise, enter your answer in the form ( x , y ).

Accepted Answer

The answer of Use Cramer s rule to find the...

Question 13

Solve the system by the substitution or elimination method. $$\left( \begin{array} { r l } 
x ^ { 2 } + y ^ { 2 } & = 26 \
x - y & = 8
\end{array} \right)$$ Express your answer as an ordered pair. If there is more than one solution, separate your answers with a comma.

Accepted Answer

The answer of Solve the system by the substitution or...

Question 14

Solve the system by the substitution or elimination method. $$\left( \begin{array} { r l r } 
x ^ { 2 } + 4 y ^ { 2 } & = 25 \
x y & = 6
\end{array} \right)$$

Accepted Answer

A)  $\left( 6 , \frac { 3 } { 2 } \right) , \left( - 6 , - \frac { 3 } { 2 } \right) , ( 3,2 ) , ( - 3 , - 2 )$ 
B)  $\left( 4 , \frac { 3 } { 2 } \right) , \left( - 4 , - \frac { 3 } { 2 } \right) , ( 3,2 ) , ( - 3 , - 2 )$ 
C)  $\left( \frac { 3 } { 2 } , 4 \right) , \left( - \frac { 3 } { 2 } , - 4 \right) , ( 2,3 ) , ( - 2 , - 3 )$ 
D)  $\left( 4 , \frac { 3 } { 2 } \right) , \left( - 4 , - \frac { 3 } { 2 } \right) , ( 2,3 ) , ( - 2 , - 3 )$ 
E)  $\left( \frac { 3 } { 2 } , 4 \right) , \left( - \frac { 3 } { 2 } , - 4 \right) , ( 3,2 ) , ( - 3 , - 2 )$ 
A)  $\left( 6 , \frac { 3 } { 2 } \right) , \left( - 6 , - \frac { 3 } { 2 } \right) , ( 3,2 ) , ( - 3 , - 2 )$ 
B)  $\left( 4 , \frac { 3 } { 2 } \right) , \left( - 4 , - \frac { 3 } { 2 } \right) , ( 3,2 ) , ( - 3 , - 2 )$ 
C)  $\left( \frac { 3 } { 2 } , 4 \right) , \left( - \frac { 3 } { 2 } , - 4 \right) , ( 2,3 ) , ( - 2 , - 3 )$ 
D)  $\left( 4 , \frac { 3 } { 2 } \right) , \left( - 4 , - \frac { 3 } { 2 } \right) , ( 2,3 ) , ( - 2 , - 3 )$ 
E)  $\left( \frac { 3 } { 2 } , 4 \right) , \left( - \frac { 3 } { 2 } , - 4 \right) , ( 3,2 ) , ( - 3 , - 2 )$

Question 15

Evaluate $4 \times 4$ determinant. $$\left| \begin{array} { c c c c } 
1 & - 3 & 4 & 3 \
3 & - 1 & 0 & 6 \
- 4 & 6 & 0 & - 3 \
- 1 & 1 & 1 & 8
\end{array} \right|$$ Use the properties of determinants to your advantage.

Accepted Answer

A)  330 
B)  250 
C)  466 
A)  330 
B)  250 
C)  466

Question 16

Solve the system. $$\left( \begin{array} { r l } 
3 x - 5 y - 2 z & = 24 \
x - y + z & = 7 \
- 3 x + 4 y - 2 z & = - 24
\end{array} \right)$$

Accepted Answer

A)  x = -1, y = -6, z = 2 
B)  x = -1, y = -7, z = 2 
C)  x = -4, y = -3, z = -1 
D)  x = 2, y = -4, z = 1 
A)  x = -1, y = -6, z = 2 
B)  x = -1, y = -7, z = 2 
C)  x = -4, y = -3, z = -1 
D)  x = 2, y = -4, z = 1

Question 17

Use the appropriate property of determinants from this section to justify the true statement. $$( - 5 ) \left| \begin{array} { c c c } 
3 & 1 & - 1 \
7 & 3 & 1 \
3 & 1 & 7
\end{array} \right| = \left| \begin{array} { c c c } 
3 & - 5 & - 1 \
7 & - 15 & 1 \
3 & - 5 & 7
\end{array} \right|$$ Do not evaluate the determinants.

Accepted Answer

A)  Property number 4: If square matrix B is obtained from square matrix A by adding k times a row (or column) of A to another row (or column) of A , then | B | = | A |. 
B)  Property number 2: If square matrix B is obtained from square matrix A by interchanging two rows (or two columns), then | B | = $-$ | A |. 
C)  Property number 1: If any row (or column) of a square matrix A contains only zeros, then | A | = 0. 
D)  Property number 5: If two rows (or columns) of a square matrix A are identical, then | A | = 0. 
E)  Property number 3: If square matrix B is obtained from square matrix A by multiplying each element of any row (or column) of A by some real number k , then | B | = k | A |. 
A)  Property number 4: If square matrix B is obtained from square matrix A by adding k times a row (or column) of A to another row (or column) of A , then | B | = | A |. 
B)  Property number 2: If square matrix B is obtained from square matrix A by interchanging two rows (or two columns), then | B | = $-$ | A |. 
C)  Property number 1: If any row (or column) of a square matrix A contains only zeros, then | A | = 0. 
D)  Property number 5: If two rows (or columns) of a square matrix A are identical, then | A | = 0. 
E)  Property number 3: If square matrix B is obtained from square matrix A by multiplying each element of any row (or column) of A by some real number k , then | B | = k | A |.

Question 18

Evaluate $2 \times 2$ determinant. $$\left| \begin{array} { c c } 
4 & - 7 \
9 & - 4
\end{array} \right|$$

Accepted Answer

A)  47 
B)  59 
C)  53 
A)  47 
B)  59 
C)  53

Question 19

Evaluate $4 \times 4$ determinant. $$\left| \begin{array} { c c c c } 
1 & 2 & 0 & 0 \
4 & - 1 & 5 & 6 \
- 2 & 5 & 1 & 8 \
2 & - 1 & - 2 & - 4
\end{array} \right|$$ Use the properties of determinants to your advantage.

Accepted Answer

A)  -186 
B)  -206 
C)  -358 
A)  -186 
B)  -206 
C)  -358

Question 20

Graph the system so that approximate real number solutions (if there are any) can be predicted. $$\left( \begin{array} { r l } 
x ^ { 2 } - 4 y ^ { 2 } & = 16 \
2 y - x & = 2
\end{array} \right)$$

Accepted Answer

A)    
B)    
C)    
D)    
E)    
A)    
B)    
C)    
D)    
E)

Solve the system. $\left(\begin{array}{rl}4 x+3 y-4 z & =-15 \\3 x-3 y+4 z & =43 \\5 y+z & =-21\end{array}\right)$

A video store rents new release movies for $5 and favorites for $2.55. One day the number of new release movies rented was twice the number of favorites. If the total income for that day was $815.75, how many movies of each kind were rented?

The augmented matrix of a system of equations is a matrix of the coefficients of the equations.

Match each step with step-by-step description of how you would solve the system. $\left( \begin{array} { c } 2 x - y + 3 z = 24 \\x - 2 y - z = 1 \\3 x + 5 y + 8 z = 47\end{array} \right)$ -Finding x , y , z . x = 4, y = -1 , z = 5.

Evaluate $2 \times 2$ determinant. $\left|\begin{array} { c c } \frac { 1 } { 4 } & \frac { 4 } { 15 } \\\frac { 15 } { 16 } & - \frac { 1 } { 15 }\end{array} \right|$

Solve the system by the substitution or elimination method. $\left( \begin{array} { r l } 7 x ^ { 2 } + y ^ { 2 } & = 343 \\x ^ { 2 } + y ^ { 2 } & = 49\end{array} \right)$

Graphs of two hyperbolas will have have no points of intersection.

The matrix $\left( \begin{array} { l } 3 x + y - z = 6 \\2 y + 3 z = 4 \\4 z = 10\end{array} \right)$ is in echelon form.

Solve the system. $\left( \begin{array} { r l } x + 2 y - z & = 9 \\x + 3 y + 2 z & = 11 \\- x - 3 y - 2 z & = - 11\end{array} \right)$

Solve the system by the substitution or elimination method. $\left( \begin{array} { r l } x ^ { 2 } + y ^ { 2 } & = 26 \\x - y & = 8\end{array} \right)$ Express your answer as an ordered pair. If there is more than one solution, separate your answers with a comma.

Solve the system by the substitution or elimination method. $\left( \begin{array} { r l r } x ^ { 2 } + 4 y ^ { 2 } & = 25 \\x y & = 6\end{array} \right)$

Evaluate $4 \times 4$ determinant. $\left| \begin{array} { c c c c } 1 & - 3 & 4 & 3 \\3 & - 1 & 0 & 6 \\- 4 & 6 & 0 & - 3 \\- 1 & 1 & 1 & 8\end{array} \right|$ Use the properties of determinants to your advantage.

Solve the system. $\left( \begin{array} { r l } 3 x - 5 y - 2 z & = 24 \\x - y + z & = 7 \\- 3 x + 4 y - 2 z & = - 24\end{array} \right)$

Evaluate $2 \times 2$ determinant. $\left| \begin{array} { c c } 4 & - 7 \\9 & - 4\end{array} \right|$

Evaluate $4 \times 4$ determinant. $\left| \begin{array} { c c c c } 1 & 2 & 0 & 0 \\4 & - 1 & 5 & 6 \\- 2 & 5 & 1 & 8 \\2 & - 1 & - 2 & - 4\end{array} \right|$ Use the properties of determinants to your advantage.

Graph the system so that approximate real number solutions (if there are any) can be predicted. $\left( \begin{array} { r l } x ^ { 2 } - 4 y ^ { 2 } & = 16 \\2 y - x & = 2\end{array} \right)$

Some Basic Concepts of Arithmetic and Algebra

The Real Numbers

Equations, Inequalities and Problem Solving

Formulas and Problem Solving

Coordinate Geometry and Linear Systems

Exponents and Polynomials

Factoring, Solving Equations and Problem Solving

A Transition From Elementary Algebra to Intermediate Algebra

Rational Expressions

Exponents and Radicals

Quadratic Equations and Inequalities

Coordinate Geometry: Lines, Parabolas, Circles, Ellipses and Hyperbolas

Functions

Exponential and Logarithmic Functions

Filters

Exam 15: Systems of Equations: Matrices and Determinants

Solve the system. (4x+3y−4z=−153x−3y+4z=435y+z=−21)\left(\begin{array}{rl}4 x+3 y-4 z & =-15 \\3 x-3 y+4 z & =43 \\5 y+z & =-21\end{array}\right)​4x+3y−4z3x−3y+4z5y+z​=−15=43=−21​​

A video store rents new release movies for $5 and favorites for $2.55. One day the number of new release movies rented was twice the number of favorites. If the total income for that day was $815.75, how many movies of each kind were rented?

The augmented matrix of a system of equations is a matrix of the coefficients of the equations.

Evaluate 2×22 \times 22×2 determinant. ∣144151516−115∣\left|\begin{array} { c c } \frac { 1 } { 4 } & \frac { 4 } { 15 } \\\frac { 15 } { 16 } & - \frac { 1 } { 15 }\end{array} \right|​41​1615​​154​−151​​​

Solve the system by the substitution or elimination method. (7x2+y2=343x2+y2=49)\left( \begin{array} { r l } 7 x ^ { 2 } + y ^ { 2 } & = 343 \\x ^ { 2 } + y ^ { 2 } & = 49\end{array} \right)(7x2+y2x2+y2​=343=49​)

Graphs of two hyperbolas will have have no points of intersection.

The matrix (3x+y−z=62y+3z=44z=10)\left( \begin{array} { l } 3 x + y - z = 6 \\2 y + 3 z = 4 \\4 z = 10\end{array} \right)​3x+y−z=62y+3z=44z=10​​ is in echelon form.

Solve the system. (x+2y−z=9x+3y+2z=11−x−3y−2z=−11)\left( \begin{array} { r l } x + 2 y - z & = 9 \\x + 3 y + 2 z & = 11 \\- x - 3 y - 2 z & = - 11\end{array} \right)​x+2y−zx+3y+2z−x−3y−2z​=9=11=−11​​

Solve the system by the substitution or elimination method. (x2+4y2=25xy=6)\left( \begin{array} { r l r } x ^ { 2 } + 4 y ^ { 2 } & = 25 \\x y & = 6\end{array} \right)(x2+4y2xy​=25=6​)

Solve the system. (3x−5y−2z=24x−y+z=7−3x+4y−2z=−24)\left( \begin{array} { r l } 3 x - 5 y - 2 z & = 24 \\x - y + z & = 7 \\- 3 x + 4 y - 2 z & = - 24\end{array} \right)​3x−5y−2zx−y+z−3x+4y−2z​=24=7=−24​​

Evaluate 2×22 \times 22×2 determinant. ∣4−79−4∣\left| \begin{array} { c c } 4 & - 7 \\9 & - 4\end{array} \right|​49​−7−4​​

Graph the system so that approximate real number solutions (if there are any) can be predicted. (x2−4y2=162y−x=2)\left( \begin{array} { r l } x ^ { 2 } - 4 y ^ { 2 } & = 16 \\2 y - x & = 2\end{array} \right)(x2−4y22y−x​=16=2​)

Some Basic Concepts of Arithmetic and Algebra

The Real Numbers

Equations, Inequalities and Problem Solving

Formulas and Problem Solving

Coordinate Geometry and Linear Systems

Exponents and Polynomials

Factoring, Solving Equations and Problem Solving

A Transition From Elementary Algebra to Intermediate Algebra

Rational Expressions

Exponents and Radicals

Quadratic Equations and Inequalities

Coordinate Geometry: Lines, Parabolas, Circles, Ellipses and Hyperbolas

Functions

Exponential and Logarithmic Functions

Filters

Solve the system. $\left(\begin{array}{rl}4 x+3 y-4 z & =-15 \\3 x-3 y+4 z & =43 \\5 y+z & =-21\end{array}\right)$

Evaluate $2 \times 2$ determinant. $\left|\begin{array} { c c } \frac { 1 } { 4 } & \frac { 4 } { 15 } \\\frac { 15 } { 16 } & - \frac { 1 } { 15 }\end{array} \right|$

Solve the system by the substitution or elimination method. $\left( \begin{array} { r l } 7 x ^ { 2 } + y ^ { 2 } & = 343 \\x ^ { 2 } + y ^ { 2 } & = 49\end{array} \right)$

The matrix $\left( \begin{array} { l } 3 x + y - z = 6 \\2 y + 3 z = 4 \\4 z = 10\end{array} \right)$ is in echelon form.

Solve the system. $\left( \begin{array} { r l } x + 2 y - z & = 9 \\x + 3 y + 2 z & = 11 \\- x - 3 y - 2 z & = - 11\end{array} \right)$

Solve the system by the substitution or elimination method. $\left( \begin{array} { r l r } x ^ { 2 } + 4 y ^ { 2 } & = 25 \\x y & = 6\end{array} \right)$

Solve the system. $\left( \begin{array} { r l } 3 x - 5 y - 2 z & = 24 \\x - y + z & = 7 \\- 3 x + 4 y - 2 z & = - 24\end{array} \right)$

Evaluate $2 \times 2$ determinant. $\left| \begin{array} { c c } 4 & - 7 \\9 & - 4\end{array} \right|$

Graph the system so that approximate real number solutions (if there are any) can be predicted. $\left( \begin{array} { r l } x ^ { 2 } - 4 y ^ { 2 } & = 16 \\2 y - x & = 2\end{array} \right)$